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Properties

Label 2-495-45.4-c1-0-38
Degree $2$
Conductor $495$
Sign $-0.442 + 0.896i$
Analytic cond. $3.95259$
Root an. cond. $1.98811$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.454 − 0.262i)2-s + (−1.11 + 1.32i)3-s + (−0.862 + 1.49i)4-s + (−2.14 − 0.616i)5-s + (−0.159 + 0.894i)6-s + (0.133 − 0.0768i)7-s + 1.95i·8-s + (−0.506 − 2.95i)9-s + (−1.13 + 0.283i)10-s + (0.5 + 0.866i)11-s + (−1.01 − 2.80i)12-s + (−3.09 − 1.78i)13-s + (0.0403 − 0.0698i)14-s + (3.21 − 2.15i)15-s + (−1.21 − 2.09i)16-s − 4.15i·17-s + ⋯
L(s)  = 1  + (0.321 − 0.185i)2-s + (−0.644 + 0.764i)3-s + (−0.431 + 0.746i)4-s + (−0.961 − 0.275i)5-s + (−0.0653 + 0.365i)6-s + (0.0503 − 0.0290i)7-s + 0.690i·8-s + (−0.168 − 0.985i)9-s + (−0.359 + 0.0898i)10-s + (0.150 + 0.261i)11-s + (−0.292 − 0.811i)12-s + (−0.857 − 0.494i)13-s + (0.0107 − 0.0186i)14-s + (0.830 − 0.557i)15-s + (−0.303 − 0.524i)16-s − 1.00i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.442 + 0.896i$
Analytic conductor: \(3.95259\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1/2),\ -0.442 + 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.101250 - 0.162890i\)
\(L(\frac12)\) \(\approx\) \(0.101250 - 0.162890i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.11 - 1.32i)T \)
5 \( 1 + (2.14 + 0.616i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-0.454 + 0.262i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-0.133 + 0.0768i)T + (3.5 - 6.06i)T^{2} \)
13 \( 1 + (3.09 + 1.78i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.15iT - 17T^{2} \)
19 \( 1 - 4.91T + 19T^{2} \)
23 \( 1 + (4.22 + 2.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.11 + 8.85i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.56 - 7.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.68iT - 37T^{2} \)
41 \( 1 + (-3.30 + 5.72i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.96 - 2.86i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.52 - 3.18i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.44iT - 53T^{2} \)
59 \( 1 + (-3.08 + 5.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.845 + 1.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.12 + 3.53i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 16.5iT - 73T^{2} \)
79 \( 1 + (1.18 + 2.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-12.3 + 7.13i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.56T + 89T^{2} \)
97 \( 1 + (0.350 - 0.202i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.90386982119545049674236786063, −9.692117316538243318689392993588, −9.051394279107427371538253236461, −7.84767336322038503263782007378, −7.22077492135836057561991263241, −5.55139451089822926243959317938, −4.74339225888550616643814118862, −3.97900954269131116319317585751, −2.97830535221872667181295925439, −0.11501324619519461440406518861, 1.61361899800584084118710452311, 3.57709887275798112511632128813, 4.76624505733883706914973947965, 5.63479943024967846840765009917, 6.62417648157536934698366370660, 7.40869864662000940744546341042, 8.322686399312601515766166306077, 9.569788981025790720975441549955, 10.49008429640017787961451659726, 11.44234203854724531369010175508

Graph of the $Z$-function along the critical line